Here \(\lbrace 1^{\ast }, c^{\ast }, x^{\ast }, (cx)^{\ast }\rbrace \)  denote the dual
basis of \(\lbrace 1,c,x,cx\rbrace \) . Let \(T=1^{\ast }-c^{\ast },~P=x^{\ast }+(cx)^{\ast },~TP=x^{\ast }-(cx)^{\ast }\) , we
get another basis \(\lbrace 1,T,P,TP \rbrace \)  of \( H_{4}^{\ast }\) .
Recall from [1]} if \(A\)  is the subalgebra \(k[x]\)  of \(H_4\) , it is shown that \(A\)  is a right partial \(H_4\) -comodule algebra with the coaction
\(\rho (1)=\frac{1}{2}(1\otimes 1+1\otimes c+1\otimes cx), ~\rho ^r(x)=\frac{1}{2}(x\otimes 1+x\otimes c+x\otimes cx).\) 
